Global preservation of the doubly irreducible holonomy condition by the $\mathbb C^*$-action

Determine whether the $\mathbb C^*$-action on the space of regular minimal-immersion data preserves the condition that the holonomy representation is doubly irreducible, thereby establishing whether the action is globally defined on $\mathrm M^*(S)$ and on the associated quotient $\mathbb M_0(S)$.

Background

The paper defines a natural C∗\mathbb C^*-action on the larger space M(S)\mathrm M(S) by scaling q1q_1 by ζ\zeta and q2‾\overline{q_2} by ζ−1\zeta^{-1} while leaving the complex structures and the conformal factor unchanged. This action preserves the Gauss equation and infinitesimal rigidity.

The restricted space M∗(S)\mathrm M^*(S) additionally requires the holonomy to lie in the open subset of doubly irreducible representations. The authors prove only a local version of the action near each point, which is sufficient to define an infinitesimal generator and derive Hamiltonian properties for the complex area functional. Whether the action preserves the holonomy condition globally remains unresolved in the paper.

References

Unfortunately, it does not seem clear whether the action of $\mathbb C*$ preserves the condition of having holonomy within $\chi*(\pi_1(S), PSL(2,)2)$, hence we cannot say that $\mathbb C*$ acts on $\mathrm M*(S)$.

— Extending Donaldson's HyperKähler Structure via Minimal Surfaces into $\mathrm{PSL}(2,\mathbb{C})$  (2610.22288 - Emam et al., 15 Oct 2026) in Section 7.1, subsection “The infinitesimal $\mathbb C^*$-action”